This invention relates to the computation of high-resolution Radon transforms used in the processing of seismic data.
Radon transforms, along with Fourier and several other transforms, are part of the tools available to geophysicists for modeling and analysis of seismic signals. In geophysics and in other application areas, improved Radon transforms have been obtained by overcoming limitations due to the sampling and noise content of the data.
The earliest Radon transforms used in geophysics are simply discrete versions of results derived for continuous functions sampled over a large interval. The discrete Radon transform algorithms introduced later (Beylkin, 1987, Hampson, 1986) provide exactly invertible transforms for discretely sampled, aperture-limited data. These transforms, now referred to as conventional, are effective at separating signal and noise provided that the input data and their line integrals (i.e., data in the Radon transform domain) are sampled without aliasing. In practice, processing of aliased-input data is an important issue, and improvements to current practice have been pursued by two approaches.
The first approach consists of deriving sampling requirements on conventional Radon transforms (Schonewille and Duijndam, 2001, Hugonnet and Canadas, 1995, Marfurt, 1996), and meeting these requirements through data acquisition, data interpolation, and pre-processing (Manin and Spitz, 1995).
In the second approach, prior information about the parts of the data that are aliased is included in the computation of the Radon transform. In its simplest form, prior information is introduced as a diagonal regularization term in the least-squares solution for the Radon transform. When all values along the diagonal are equal, the resulting solutions do not in general resolve ambiguities due to aliased input data. Nichols (1992a, 1992b), Herrmann et al. (2000), and Hugonnet (2001) have demonstrated that appropriate diagonal regularization produce accurate and efficient Radon transforms even when the input data are aliased.
Within the framework of Bayesian estimation, the diagonal regularization can be interpreted as assigning Gaussian prior distributions to the transform parameters, which are also assumed to be statistically independent (Tarantola, 1987, Ulrych et al., 2001). Other algorithms require less prior information by avoiding the assumption of Gaussian distributions, but tend to be computationally more complex (Sacchi and Ulrych, 1995, Harlan et al., 1984, Thorson and Claerbout, 1985).
Prior art on diagonal regularization in the computation of the high-resolution Radon transforms include the methods described by Nichols (1992a), Herrmann et al. (2000), and Hugonnet et al. (2001). In these cases, for reasons of computational efficiency, the transform is usually applied in the space-frequency domain with frequency-dependent diagonal regularization. The regularization term, however, is not updated as a function of the computed Radon transform.
Nichols (1992a) derives the regularization weights from a semblance measure along the temporal frequency axis of the data. The semblance for non-aliased seismic energy is usually varying slowly with frequency. Conversely, strong, short scale variations in the semblance are often used to identify aliased energy. To derive the weights, Nichols (1992a) discloses the smoothing of the semblance over a frequency interval containing frequencies that are lower and higher than the frequency being processed.
Herrmann (2000) and Hugonnet (2001) derive weights recursively in frequency, starting from a conventional Radon transform at the lowest frequency. The weights for a frequency being processed are built from the results of the high-resolution Radon transform at lower frequencies.
The examples in FIGS. 2, 3, and 4 illustrate Radon transforms computed with different prior art regularization schemes. FIGS. 2A-2D correspond to a conventional Radon transform regularized with a scalar multiplied by the identity matrix. In FIGS. 3A-3D, the regularization weights are variable terms along the diagonal, and the corresponding solution has higher resolution. In FIG. 4 however, we see that the same regularization scheme applied to noise-contaminated data produces a sub-optimal result.
The weights in FIG. 4E do not detect the parabolic events at low and high frequencies. The weights indicate reliably the moveouts in the data only above 15 Hz, while in the noise-free case as shown in FIG. 3E, the moveouts are detected from about 5 Hz. The weights in FIG. 4E also have spuriously high values at the edges of the transform domain, possibly an artifact due to starting the transform from a conventional Radon transform (i.e., equal weights). These high values generate strong artifacts in the data domain.
A new method for deriving regularization weights for the computation of high-resolution Radon transforms is described.
In one embodiment of the invention, a method of processing seismic data is presented. A high-resolution Radon transform is defined for use on seismic data. The high-resolution Radon transform is regularized using a semblance measure of the seismic data. The seismic data is processed using the high-resolution Radon transform to enhance desirable features of the seismic data. A tangible representation of the processed seismic data is presented.
The semblance measure of the seismic can include a semblance measure along a dimension of the seismic data. The seismic data can be divided into a two-dimensional array, which includes one dimension of time or depth, and a second dimension of a spatial surface position or an angle. The seismic data can also be divided into a multi-dimensional array including one dimension including time or depth, and the other dimensions selected from spatial surface positions and angles.
The processing of the seismic data can include performing the high-resolution Radon transform on the seismic data using the semblance measure of the seismic data. The first region and a second region in the Radon transformed seismic data can be separated. An inverse of the high-resolution Radon transform on the separated, Radon transformed seismic data can be performed.
The first region can be a signal region and the second region can be a noise region. The dimensions of the seismic data can include multiple dimensions of the seismic data. The dimension of the seismic data can also include a frequency domain.
In another embodiment, the processing of the seismic data using the high-resolution Radon transform to enhance desirable features of the seismic data can further include approximating a complex moveout trajectory by segments. The seismic data can be divided into local windows of data consistent with the segments. The high-resolution Radon transform can be performed on each of the local windows of data to enhance the desirable features of the seismic data consistent with the segments. Performing the high-resolution Radon transform on each of the local windows of data can include the steps of computing the transform with periodic boundary conditions and applying the signal-and-noise separation consistent with zero-value boundary conditions by using the known timeshifts between components of the model.
Regularizing the high-resolution Radon transform using a semblance measure of the seismic data can include computing a regularization matrix, which includes applying a phase shift determined by a moveout trajectory to the seismic data. It can also include normalizing the stack power along the moveout trajectory to obtain a semblance measure. It can also include sharpening the semblance measure or smoothing the semblance measure over a second dimension of the seismic data. The second dimension of the seismic data can include a dimension selected from a spatial position and an angle.
In another embodiment, the invention can be implemented on a computer system, where the computer includes a memory and a processor, and executable software residing in the computer memory. The software is operative with the processor to define a high-resolution Radon transform for use on seismic data. The high-resolution Radon transform is regularized using a semblance measure of the seismic data. The seismic data is processed using the high-resolution Radon transform to enhance desirable features of the seismic data.
The software can also provide for the performance of the high-resolution Radon transform on the seismic data using the semblance measure of the seismic data. A first region and a second region can be separated in the Radon transformed seismic data. An inverse of the high-resolution Radon transform on the separated, Radon transformed seismic data can be used.
The software can also include approximating a complex moveout trajectory by segments. The seismic data can be divided into a local windows of data consistent with the segments. A high-resolution Radon transform can be performed on each of the local windows of data to enhance the desirable features of the seismic data consistent with the plurality of segments.
The proposed method may retain the resolution and dealiasing qualities associated with high-resolution Radon transforms, and most of the efficiency associated with Nichols"" and Herrmann""s methods, but can be more stable than Herrrnann""s method and can provide better resolution than Nichols"" method.
Other features and advantages of the invention will be further understood in view of the following description.